
\chapter*{Tables of Notation}

%\section*{Technical Notation}
%\begin{table}

\begin{center}
\begin{tabular}{|ll|}
\hline
\(\mathcal{X,Y,Z}\) & sets of  objects \\
\(X,Y,Z\) & random variables   \\
\(x,y,z\) & (  objects \\
\({\bf x,y,z}\) & vectors in high-dimensional space \\
%\(P\) & a probability mass function \\
\(f: \mathcal{X}\rightarrow \mathcal{Y}\)  & a function from input domain \(\mathcal{X}\) to output range \(\mathcal{Y}\)\\ 
\(\max f\) & the highest value of \(f\) \\
\(\min f\) &  the lowest value of \(f\)  \\
\(\argmax f\) & the argument obtaining the highest value of \(f\) \\ 
\(\argmin f\) &  the argument obtaining the lowest value of \(f\) \\ 
\(O(n)\) & the complexity of an algorithm applied to input string of length \(n\)\\
\(\expo(x), e^x\) & the exponential function \\
\(\log(x)\) & the log function \\
\(\ffn(x,y) \) & a feature function in a high dimensional space\\
 \(\wv\) & a parametrized weight vector in  a high dimensional space\\ 
 \(\omega\in\wv\) & a  weight-vector assignment \\
 \(M\) & a statistical model \\
\(m\in M\) & a model instance \\
 \(\mathcal{H}\) & a hypothesis class \\
\(h\in\mathcal{H}\) & a prediction function\\
\data & a finite corpus \\
\(\#(x,\data)\) & the frequency of \(x\) in \data~  (or \(\#(x)\) when unambiguous)  \\
\(| {\bf x}|\) & the size of a vector \\
\(||{\bf x}||\) & the norm of a vector \\
\(P(X=x)\) & the probability that \(X=x\), abbreviated  \(P(x)\)\\
\(P(X=x|Y=y)\) & the probability that \(X=x\) given that \(Y=y\), abbreviate \(P(x|y)\)\\
\(\hat{P}(X=x)\) & estimated probability of \(X=x\), abbreviated  \(\hat{P}(x)\)\\
\(\hat{P}(X=x|Y=y)\) & estimated probability of \(X=x\) given that \(Y=y\), abbreviate \(\hat{P}(x|y)\)\\

%\(\circ\) &  composition \\
\hline
\end{tabular}
%\caption{Mathematical Notation}\label{math}
%\end{table}
\end{center} 

%\section*{Linguistic Notation}


\begin{center}
\begin{tabular}{|ll|}
\hline
 \(\Sigma\) & a finite alphabet or a finite lexicon  \\
% \(w\in Sigma\) & a word \\
\(\mctok\) & a set of tokens \\
\(\mcl\) & a set of lemmas \\
\(\mco\) & a set of operations \\
\mcc & a set of lexical (part-of-speech) categories \\
\mcn & a set of syntactic (or, phrase types) categories \\
\mcr & a set of grammatical relations (or, dependency types) labels \\
\mca & a set of attribute names \\
\mcv & a set of attribute values \\
\mcva & the range of attribute values for an attribute  \(a\in\mca\) \\
\(a:v\) &  a linguistic property  \\
\(\mcb \) & a linguistic-properties dictionary \( \{ a:v | v\in V_a\}\)  \\
\(b\subset \mcb\) &  a linguistic-properties bundle, contained in \mcb  \\
\(e\in\Sigma \) & a lexical entry \\
\(\rho(l) \) & a paradigm for \(l\in\mcl\) \\
\ma & a morphological analysis function \\
\(\mcs\) & a set of  spell-out   possibilities  (a.k.a segmentation/tokenization)\\

\(L\) & a language \\
\(T\) & a sequence of space delimited-tokens \\
\(N\) & a set of non-terminal nodes \\
\(A\) & a set of arcs \\
\(G(\Sigma, \mcn)\) & a grammar (when unabigous, only \(G\))\\
\(G(V,A)\) & a graph (when unabigous, only \(G\))\\
%\(V\) & a set of vertexes  \\
%\(v_o\in V\) & a set of vertexes  \\
\(\gamma(T)\in\ma(T)\) & a morphosyntactic disambiguation function\\
\(\pi(T,N)\) & a phrase-structure tree in English  \\
\(\pi(\gamma(T),N)\) & a phrase-structure tree in an MRL \\
\(\tau(T,A)\) & a dependency tree in English  \\
\(\tau(\gamma(T),A)\) & a dependency tree in an MRL \\
\buff & a buffer \\ 
\stack & a stack \\
& a configuration \\
& a transition \\
& an Oracle function \\
\(\circ\)& compose \\
\(\rightarrow\) & rewrite \\
\(\Rightarrow\) & derive \\
& fuse \\
\hline
\end{tabular}
%\caption{Linguistic Notation}\label{math}
\end{center} 
